Approximating discrete valuation rings by regular local rings
Authors:
William Heinzer, Christel Rotthaus and Sylvia Wiegand
Journal:
Proc. Amer. Math. Soc. 129 (2001), 3743
MSC (1991):
Primary 13F30, 13H05; Secondary 13E05, 13G05, 13J05, 13J15
Published electronically:
July 27, 2000
MathSciNet review:
1694346
Fulltext PDF Free Access
Abstract 
References 
Similar Articles 
Additional Information
Abstract: Let be a field of characteristic zero and let be a discrete rankone valuation domain containing with . Assume that the fraction field of has finite transcendence degree over . For every positive integer , we prove that can be realized as a directed union of regular local subalgebras of of dimension .
 [A]
Shreeram
Abhyankar, On the valuations centered in a local domain, Amer.
J. Math. 78 (1956), 321–348. MR 0082477
(18,556b)
 [AH]
Shreeram
S. Abhyankar and William
J. Heinzer, Ramification in infinite integral extensions, J.
Algebra 170 (1994), no. 3, 861–879. MR 1305267
(95k:13006), http://dx.doi.org/10.1006/jabr.1994.1367
 [C1]
Steven
Dale Cutkosky, Local factorization of birational maps, Adv.
Math. 132 (1997), no. 2, 167–315. MR 1491444
(99c:14018), http://dx.doi.org/10.1006/aima.1997.1675
 [C2]
, Local factorization and monomialization of morphisms, to appear.
 [HRS]
William
Heinzer, Christel
Rotthaus, and Judith
D. Sally, Formal fibers and birational extensions, Nagoya
Math. J. 131 (1993), 1–38. MR 1238631
(95a:13008)
 [HRW1]
William
Heinzer, Christel
Rotthaus, and Sylvia
Wiegand, Noetherian rings between a semilocal domain and its
completion, J. Algebra 198 (1997), no. 2,
627–655. MR 1489916
(99a:13008), http://dx.doi.org/10.1006/jabr.1997.7169
 [HRW2]
William
Heinzer, Christel
Rotthaus, and Sylvia
Wiegand, Building Noetherian domains inside an idealadic
completion, Abelian groups, module theory, and topology (Padua, 1997)
Lecture Notes in Pure and Appl. Math., vol. 201, Dekker, New York,
1998, pp. 279–287. MR 1651173
(99m:13035)
 [HRW3]
, Noetherian domains inside a homomorphic image of a completion, J. Algebra 215 (1999), 666681. CMP 99:12
 [M1]
Hideyuki
Matsumura, Commutative ring theory, Cambridge Studies in
Advanced Mathematics, vol. 8, Cambridge University Press, Cambridge,
1986. Translated from the Japanese by M. Reid. MR 879273
(88h:13001)
 [M2]
Hideyuki
Matsumura, Commutative algebra, 2nd ed., Mathematics Lecture
Note Series, vol. 56, Benjamin/Cummings Publishing Co., Inc., Reading,
Mass., 1980. MR
575344 (82i:13003)
 [N1]
Masayoshi
Nagata, An example of a normal local ring which is analytically
reducible, Mem. Coll. Sci. Univ. Kyoto. Ser. A. Math.
31 (1958), 83–85. MR 0097395
(20 #3864)
 [N2]
Masayoshi
Nagata, Local rings, Interscience Tracts in Pure and Applied
Mathematics, No. 13, Interscience Publishers a division of John Wiley &
Sons New YorkLondon, 1962. MR 0155856
(27 #5790)
 [R]
Christel
Rotthaus, Homomorphic images of regular local rings, Comm.
Algebra 24 (1996), no. 2, 445–476. MR 1373487
(97b:13029), http://dx.doi.org/10.1080/00927879608825580
 [S]
David
L. Shannon, Monoidal transforms of regular local rings, Amer.
J. Math. 95 (1973), 294–320. MR 0330154
(48 #8492)
 [Z1]
Oscar
Zariski, Local uniformization on algebraic varieties, Ann. of
Math. (2) 41 (1940), 852–896. MR 0002864
(2,124a)
 [Z2]
Oscar
Zariski, Applicazioni geometriche della teoria delle
valutazioni, Univ. Roma. Ist. Naz. Alta Mat. Rend. Mat. e Appl. (5)
13 (1954), 51–88 (Italian). MR 0063704
(16,165f)
 [ZS]
Oscar
Zariski and Pierre
Samuel, Commutative algebra. Vol. II, The University Series in
Higher Mathematics, D. Van Nostrand Co., Inc., Princeton, N.
J.TorontoLondonNew York, 1960. MR 0120249
(22 #11006)
 [A]
 S. Abhyankar, On the valuations centered in a local domain, Amer. J. Math. 78 (1956), 321348. MR 18:556b
 [AH]
 S. Abhyankar and W. Heinzer, Ramification in infinite integral extensions, J. Algebra 170 (1994), 861879. MR 95k:13006
 [C1]
 S. Cutkosky, Local factorization of birational maps, Adv. Math. 132 (1997), 167315. MR 99c:14018
 [C2]
 , Local factorization and monomialization of morphisms, to appear.
 [HRS]
 W. Heinzer, C. Rotthaus and J. Sally, Formal fibers and birational extensions, Nagoya Math J. 131 (1993), 138. MR 95a:13008
 [HRW1]
 W. Heinzer, C. Rotthaus and S. Wiegand, Noetherian rings between a semilocal domain and its completion, J. Algebra 198 (1997), 627655. MR 99a:13008
 [HRW2]
 , Building Noetherian domains inside an idealadic completion, Abelian Groups, Module Theory and Topology Proceedings in Honor of Adalberto Orsatti's 60th Birthday (D. Dikranjan and L. Salce, ed.), Dekker Inc, 1998, pp. 279287. MR 99m:13035
 [HRW3]
 , Noetherian domains inside a homomorphic image of a completion, J. Algebra 215 (1999), 666681. CMP 99:12
 [M1]
 H. Matsumura, Commutative Ring Theory, Cambridge University Press, Cambridge, 1986. MR 88h:13001
 [M2]
 , Commutative Algebra, second edition, Benjamin/Cummings, Reading, MA, 1980. MR 82i:13003
 [N1]
 M. Nagata, An example of a normal local ring which is analytically reducible, Mem. Coll. Sci., Univ. Kyoto 31 (1958), 8385. MR 20:3864
 [N2]
 , Local Rings, John Wiley, 1962. MR 27:5790
 [R]
 C. Rotthaus, Homomorphic images of regular local rings, Comm. in Alg. 24 (1996), 445476. MR 97b:13029
 [S]
 D. Shannon, Monoidal transforms of regular local rings, Amer. J. Math. 95 (1973), 284320. MR 48:8492
 [Z1]
 O. Zariski, Local uniformization on algebraic varieties, Ann. Math. 41 (1940), 852896. MR 2:124a
 [Z2]
 , Applicazioni geometriche della teoria delle valutazioni, Rend. Circolo Mat. Palermo 13(5) (1954), 138. MR 16:165f
 [ZS]
 O. Zariski and P. Samuel, Commutative Algebra vol II, Van Nostrand, Princeton, NJ, 1960. MR 22:11006
Similar Articles
Retrieve articles in Proceedings of the American Mathematical Society
with MSC (1991):
13F30,
13H05,
13E05,
13G05,
13J05,
13J15
Retrieve articles in all journals
with MSC (1991):
13F30,
13H05,
13E05,
13G05,
13J05,
13J15
Additional Information
William Heinzer
Affiliation:
Department of Mathematics, Purdue University, West Lafayette, Indiana 479071395
Email:
heinzer@math.purdue.edu
Christel Rotthaus
Affiliation:
Department of Mathematics, Michigan State University, East Lansing, Michigan 488241027
Email:
rotthaus@math.msu.edu
Sylvia Wiegand
Affiliation:
Department of Mathematics and Statistics, University of Nebraska, Lincoln, Nebraska 685880323
Email:
swiegand@math.unl.edu
DOI:
http://dx.doi.org/10.1090/S0002993900054927
PII:
S 00029939(00)054927
Keywords:
Discrete rankone valuation domain,
\'{e}tale extension,
excellent ring,
Henselization,
local uniformization,
regular local domain
Received by editor(s):
July 23, 1998
Received by editor(s) in revised form:
March 22, 1999
Published electronically:
July 27, 2000
Additional Notes:
The authors thank the National Science Foundation and the National Security Agency for support for this research. In addition they are grateful for the hospitality and cooperation of Michigan State University, the University of Nebraska and Purdue University, where several work sessions on this research were conducted.
Communicated by:
Wolmer V. Vasconcelos
Article copyright:
© Copyright 2000
American Mathematical Society
